Liquid Ellipsoid under its own Attraction. 



43 



And since %, A are not altered by interchanging the rows and 

 columns of A, we have at once Dedekind's theorem. It is 

 assumed that motions of the type (1) exist ; this was proved 

 by Dirichlet and Basset. 



Dedekind's Ellipsoid. 



The motion in Jacobi's ellipsoid, the fluid rotating as if 

 rigid, is expressed by 



x=x Q,os]ct—yQ$mkt, y=x Q &va.ht+y Q c,os,kt, z=z . 



The correlated Dedekind's form is given by 



x — x cos ht + ~ y sin ht, 



y= — x — sin Jct+y cos ht, 

 a 



(9) 



V= Tryp 



z= z . 

 The free surface is 



r 2 ?y 2 2 2 



«o 2 K c<? 



Since the motion is steady we may drop the suffix zero in 

 a , b , c . 



The Lagrangian equations are three, of the form 



O*o O*o O*o O*o P 0*o 



where p is the density, p the pressure, and Y the gravitation- 

 potential; so that 



dyfr «* " 2 



1 



\/(»SX'4X' + l) 



/ 1 __^ 2 £ *_\ 



and the pressure 



p= const. +«r/l-- 5r -- gr -^ r | 



"With the usual notation, 

 oV 



B* 



= -Aa?, . . . , 



and with the above values of a?, y, z the equations of motion 

 become 



